differential equations
University of Melbourne
MAST 10005
CALCULUS 1 LECTURE NOTES
TABLE OF CONTENTS Topic 1: Set Theory 1.0 Real Numbers 1.01 Set of natural numbers 1.02 Set of integers 1.03 Set of rational numbers 1.04 Set of real numbers
Pages 1 - 2
1.1 Set Notations and Operations 1.11 Membership 1.12 Set notation 1.13 Bounded intervals 1.14 Subsets 1.15 Unions, intersections and set complements 1.16 Cartesian products 1.17 Inequalities
Pages 3 - 12
Topic 2: Complex Numbers 2.0 Complex Numbers 2.01 Properties of complex numbers 2.02 Arithmetic of complex numbers 2.03 Complex numbers in polar form 2.04 Complex exponential form 2.05 Properties of the modulus and argument 2.06 Sketching regions in the complex plane 2.07 Powers and roots of complex numbers 2.08 Roots of complex polynomials
Pages 13 - 27
Topic 3: Functions 3.0 Functions 3.01 Basics of functions 3.02 Functional arithmetic 3.03 Reciprocal trigonometric functions 3.04 Types of functions (bijective, injective, surjective) 3.05 Inverse trigonometric functions 3.06 Domain and range of composite functions 3.07 Absolute functions
Pages 28 - 38
Topic 4: Vectors 4.0 Vectors 4.01 4.02 4.03 4.04
Pages 39 - 51 Arithmetic and properties of vectors Dot (scalar) product Vector projections Parametric curves
Topic 5: Differential Calculus 5.0 Derivatives 5.01 First derivative and differentiation rules 5.02 Derivatives and graphs 5.03 Higher order derivatives 5.04 Rectilinear motion 5.05 Implicit differentiation 5.06 Derivatives of inverse trigonometric functions 5.07 Differentiating parametric curves 5.08 Projectile motion
Pages 52 - 72
Topic 6: Integral Calculus 6.0 Integration 6.01 Integrals and the Fundamental Theorem of Calculus 6.02 Integration by substitution (linear substitution) 6.03 Integration by partial fractions 6.04 Definite integrals
Pages 73 - 84
Topic 7: Differential Equations 7.0 Differential Equations 7.01 Introduction to differential equations 7.02 Separable differential equations 7.03 Population growth models 7.04 Newton’s Law of Cooling 7.05 Models for tree growth
Pages 84 - 95
TOPIC 7 - DIFFERENTIAL EQUATIONS 7.0
Differential Equations
7.01 Introduction to Differential Equations Now that we have mastered enough integration techniques we are ready to tackle what is arguably the most important part of calculus - the study of differential equations. These are a type of equation where the unknown is a function rather than a number or a set of numbers. Differential equations arise in applications where our knowledge of a physical system is expressed in terms of the rate of change in some quantity. Differential equations arise naturally in every branch of science and many areas of social science and economic as well. Indeed, a key reason why calculus is so indispensable to quantitative disciplines is the fact that differential equations are used to construct mathematical models that are essential to our understanding of quantitative relationships in so many fields. An equation involving a variable (say x), an unknown function and the derivatives of that function y with respect to x is called an ordinary differential equation or DE. The order of the highest derivative in the differential equation is called the order of the differential equation. A differential equation is really just a function equation involving some derivatives, so to check a proposed solution to a differential equation, we substitute it into both sides and see if it works for all x-values. Examples: 1.
When one is asked to verify or prove that a given function is a solution of a differential equation, there is no need to solve the differential equation. The bad news is that a typical differential equation has infinitely many solutions, but the solutions typically have similar formulas, differing only by the presence of some constants.
Examples: 1. Find a formula for all solutions of the following DE by integrating both sides.
The general solution of a differential equation is a formula for all of the solutions of the differential equation. We can find the general solution a differential equation where f(x) is a known function by integrating both sides. This gives a constant of integration in the solution formula. Similarly, we can find the general solution of a differential equation in the form below by integrating both sides twice. This gives two constants of integration in the solution.
Fixing the values of the constants of integration gives a particular solution. In a first order differential equation in an application, we typically know the value of the solution at some point. This usually allows us to solve for the constant of integration and hence find a particular solution. For example, for the particular solution of the differential equation in Example 1 above if it is subject to the condition y = 3 when x = π/2 implies that y = sin(x) + 2. The constraint y = 3 when x = π/2 is called an initial condition or initial value. For a differential equation of order n, we need initial conditions to determine the n constants of integration in the general solution. A differential equation together with one or more initial values is called an initial value problem. Examples: 1.
Note that we have to be careful when we include our constants of integration. Simply adding them at the end will give a wrong answer. Examples: 1. Find the general solution of the following differential equation and find the particular solution where y = 1.5 when x = 0.
7.02 Separable Differential Equations As we shall see, differential equations of the form shown below where F and G are known functions, arise in many real world applications. They are called separable differential equations and they can usually be solved by a technique called separation of variables, which we now explore.
The first order differential equation’s we solved in the preceding examples were all separable differential equation’s in the special case G(y) = 1. First order differential equations of the form below are called autonomous differential equations.
We can’t just integrate both sides with respect to x because the RHS is now a function of y, instead of the variable x. However, provided G(y) does not equal 0, we can divide both sides by G(y) allowing to integrate both sides, applying integration by substitution on the LHS. Integrating both sides with respect to x, using substitution on the LHS gives:
MAST 10005
CALCULUS 1 LECTURE NOTES
TABLE OF CONTENTS Topic 1: Set Theory 1.0 Real Numbers 1.01 Set of natural numbers 1.02 Set of integers 1.03 Set of rational numbers 1.04 Set of real numbers
Pages 1 - 2
1.1 Set Notations and Operations 1.11 Membership 1.12 Set notation 1.13 Bounded intervals 1.14 Subsets 1.15 Unions, intersections and set complements 1.16 Cartesian products 1.17 Inequalities
Pages 3 - 12
Topic 2: Complex Numbers 2.0 Complex Numbers 2.01 Properties of complex numbers 2.02 Arithmetic of complex numbers 2.03 Complex numbers in polar form 2.04 Complex exponential form 2.05 Properties of the modulus and argument 2.06 Sketching regions in the complex plane 2.07 Powers and roots of complex numbers 2.08 Roots of complex polynomials
Pages 13 - 27
Topic 3: Functions 3.0 Functions 3.01 Basics of functions 3.02 Functional arithmetic 3.03 Reciprocal trigonometric functions 3.04 Types of functions (bijective, injective, surjective) 3.05 Inverse trigonometric functions 3.06 Domain and range of composite functions 3.07 Absolute functions
Pages 28 - 38
Topic 4: Vectors 4.0 Vectors 4.01 4.02 4.03 4.04
Pages 39 - 51 Arithmetic and properties of vectors Dot (scalar) product Vector projections Parametric curves
Topic 5: Differential Calculus 5.0 Derivatives 5.01 First derivative and differentiation rules 5.02 Derivatives and graphs 5.03 Higher order derivatives 5.04 Rectilinear motion 5.05 Implicit differentiation 5.06 Derivatives of inverse trigonometric functions 5.07 Differentiating parametric curves 5.08 Projectile motion
Pages 52 - 72
Topic 6: Integral Calculus 6.0 Integration 6.01 Integrals and the Fundamental Theorem of Calculus 6.02 Integration by substitution (linear substitution) 6.03 Integration by partial fractions 6.04 Definite integrals
Pages 73 - 84
Topic 7: Differential Equations 7.0 Differential Equations 7.01 Introduction to differential equations 7.02 Separable differential equations 7.03 Population growth models 7.04 Newton’s Law of Cooling 7.05 Models for tree growth
Pages 84 - 95
TOPIC 7 - DIFFERENTIAL EQUATIONS 7.0
Differential Equations
7.01 Introduction to Differential Equations Now that we have mastered enough integration techniques we are ready to tackle what is arguably the most important part of calculus - the study of differential equations. These are a type of equation where the unknown is a function rather than a number or a set of numbers. Differential equations arise in applications where our knowledge of a physical system is expressed in terms of the rate of change in some quantity. Differential equations arise naturally in every branch of science and many areas of social science and economic as well. Indeed, a key reason why calculus is so indispensable to quantitative disciplines is the fact that differential equations are used to construct mathematical models that are essential to our understanding of quantitative relationships in so many fields. An equation involving a variable (say x), an unknown function and the derivatives of that function y with respect to x is called an ordinary differential equation or DE. The order of the highest derivative in the differential equation is called the order of the differential equation. A differential equation is really just a function equation involving some derivatives, so to check a proposed solution to a differential equation, we substitute it into both sides and see if it works for all x-values. Examples: 1.
When one is asked to verify or prove that a given function is a solution of a differential equation, there is no need to solve the differential equation. The bad news is that a typical differential equation has infinitely many solutions, but the solutions typically have similar formulas, differing only by the presence of some constants.
Examples: 1. Find a formula for all solutions of the following DE by integrating both sides.
The general solution of a differential equation is a formula for all of the solutions of the differential equation. We can find the general solution a differential equation where f(x) is a known function by integrating both sides. This gives a constant of integration in the solution formula. Similarly, we can find the general solution of a differential equation in the form below by integrating both sides twice. This gives two constants of integration in the solution.
Fixing the values of the constants of integration gives a particular solution. In a first order differential equation in an application, we typically know the value of the solution at some point. This usually allows us to solve for the constant of integration and hence find a particular solution. For example, for the particular solution of the differential equation in Example 1 above if it is subject to the condition y = 3 when x = π/2 implies that y = sin(x) + 2. The constraint y = 3 when x = π/2 is called an initial condition or initial value. For a differential equation of order n, we need initial conditions to determine the n constants of integration in the general solution. A differential equation together with one or more initial values is called an initial value problem. Examples: 1.
Note that we have to be careful when we include our constants of integration. Simply adding them at the end will give a wrong answer. Examples: 1. Find the general solution of the following differential equation and find the particular solution where y = 1.5 when x = 0.
7.02 Separable Differential Equations As we shall see, differential equations of the form shown below where F and G are known functions, arise in many real world applications. They are called separable differential equations and they can usually be solved by a technique called separation of variables, which we now explore.
The first order differential equation’s we solved in the preceding examples were all separable differential equation’s in the special case G(y) = 1. First order differential equations of the form below are called autonomous differential equations.
We can’t just integrate both sides with respect to x because the RHS is now a function of y, instead of the variable x. However, provided G(y) does not equal 0, we can divide both sides by G(y) allowing to integrate both sides, applying integration by substitution on the LHS. Integrating both sides with respect to x, using substitution on the LHS gives:
